Noteworthy fractal features and transport properties of Cantor tartans

Balankin, Alexander S.; Golmankhaneh, Alireza Khalili; Patiño-Ortiz, Julián; Patiño-Ortiz, Miguel

doi:10.1016/j.physleta.2018.04.011

Cited by 36 publications

(14 citation statements)

References 43 publications

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“…As a final note, we recall that normal diffusion was already observed in several fractals, i.e. D W = 2 [39][40][41]. However, infiltration may be anomalous in those systems because the exponent n also depends on the dimension of the medium and on the dimension of the source, as shown here for the case of small localized sources and in previous works for extended sources.…”

Section: Discussionsupporting

confidence: 72%

Models of infiltration into homogeneous and fractal porous media with localized sources

Reis

Voller

2019

Phys. Rev. E

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We study a random walk infiltration (RWI) model, in homogeneous and in fractal media, with small localized sources at their boundaries. In this model, particles released at a source, maintained at a constant density value, execute unbiased random walks over a lattice; a model that represents solute infiltration by diffusion into a medium in contact with a reservoir of fixed concentration. A scaling approach shows that the infiltrated length, area, or volume evolves in time as the number of distinct sites visited by a single random walker in the same medium. This is consistent with numerical simulations of the lattice model and exact and numerical solutions of the corresponding diffusion equation. In a Sierpinski carpet, the infiltrated area is expected to evolve as t D F /D W (Alexander-Orbach relation), where DF is the fractal dimension of the medium and DW is the random walk dimension; the numerical integration of the diffusion equation supports this result with very good accuracy and improves results of lattice random walk simulations. In a Menger sponge in which DF > DW (i.e., a fractal with a dimension close to 3), a linear time increase of the infiltrated volume is theoretically predicted and confirmed numerically. Thus, no evidence of fractality can be observed in measurements of infiltrated volumes or masses in media where random walks are not recurrent, although the tracer diffusion is anomalous. We compare our findings with results for a fluid infiltration model in which the pressure head is constant at the source and the front displacement is driven by the local gradient of that head. Exact solutions in two and three dimensions and numerical results in a carpet show that this type of fluid infiltration is in the same universality class of RWI, with an equivalence between the head and the particle concentration. These results set a relation between different infiltration processes with localized sources and the recurrence properties of random walks in the same media. *

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Section: Discussionsupporting

confidence: 72%

Models of infiltration into homogeneous and fractal porous media with localized sources

Reis

Voller

2019

Phys. Rev. E

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“…Actually, problems dealing with transport phenomena are very important in science and engineering, particularly, in the study of porous media there is a great research activity both theoretical and experimental [27,33,34,[40][41][42][43][45][46][47][48][49][50][51][52][53][54].…”

Section: Discussionmentioning

confidence: 99%

Generalities on finite element discretization for fractional pressure diffusion equation in the fractal continuum

2019

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In this study we explore the application of the novel fractional calculus in fractal continuum (FCFC), together with the finite element method (FEM), in order to analize explicitly how these differential operators act in the process of discretizing the generalized fractional pressure diffusion equation for a three-dimensional anisotropic continuous fractal flow. The master finite element equation (MFEE) for arbitrary interpolation functions is obtained. As an example, MFEE for the case of a generic linear tetrahedron in $\mathbb{R}^3$ is shown. Analytic solution for the spatial variables is determined over a canonical tetrahedral finite element in global coordinates.

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“…After its introduction in the papers cited above, fractal calculus has been applied in optics, classical mechanics, electromagnetics, and quantum mechanics. Random walks on fractal sets and the corresponding mean square displacements have been modelled by fractal calculus [28][29][30][31][32][33][34][35]. Non-local analogues of fractional derivatives have been defined as a model for processes on fractal sets with memory effect [36].…”

Section: Introductionmentioning

confidence: 99%

Random Variables and Stable Distributions on Fractal Cantor Sets

Golmankhaneh

Fernandez

2019

Fractal Fract

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In this paper, we introduce the concept of fractal random variables and their related distribution functions and statistical properties. Fractal calculus is a generalisation of standard calculus which includes function with fractal support. Here we combine this emerging field of study with probability theory, defining concepts such as Shannon entropy on fractal thin Cantor-like sets. Stable distributions on fractal sets are suggested and related physical models are presented. Our work is illustrated with graphs for clarity of the results.

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Noteworthy fractal features and transport properties of Cantor tartans

Cited by 36 publications

References 43 publications

Models of infiltration into homogeneous and fractal porous media with localized sources

Models of infiltration into homogeneous and fractal porous media with localized sources

Generalities on finite element discretization for fractional pressure diffusion equation in the fractal continuum

Random Variables and Stable Distributions on Fractal Cantor Sets

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